SO(3)-invariant PCA with application to molecular data

ABSTRACT Principal component analysis (PCA) is a fundamental technique for dimensionality reduction and denoising; however, its application to three-dimensional data with arbitrary orientations--common in structural biology--presents significant challenges. A naive approach requires augmenting the dataset with many rotated copies of each sample, incurring prohibitive computational costs. In this paper, we extend PCA to 3D volumetric datasets with unknown orientations by developing an efficient and principled framework for SO(3)-invariant PCA that implicitly accounts for all rotations without explicit data augmentation. By exploiting underlying algebraic structure, we demonstrate that the computation involves only the square root of the total number of covariance entries, resulting in a substantial reduction in complexity. Index T erms-- steerable PCA, group invariants, 3D volumes, cryo-EM, spherical Bessel, ball harmonics 1. INTRODUCTION Principal component analysis (PCA) is a fundamental technique in data science and statistics, especially when dealing with high-dimensional datasets.